Distribution functions of ratio sequences. An expository paper

Abstract

This expository paper presents known results on distribution functions g(x) of the sequence of blocks where xn is an increasing sequence of positive integers. Also presents results of the set G(Xn) of all distribution functions g(x). Specially: - continuity of g(x); - connectivity of G(Xn); - singleton of G(Xn); - one-step g(x); - uniform distribution of Xn, n = 1, 2, . . . ; - lower and upper bounds of g(x); - applications to bounds of ; - many examples, e.g., , where pn is the nth prime, is uniformly distributed. The present results have been published by 25 papers of several authors between 2001-2013.

Distribution functions, extremal limits and optimal transport

“…More recently, the study of distribution functions was extended to multivariate functions by the Slovak school of O. Strauch and his coworkers; see [5,6,24,44]. In particular, they studied properties of the set of distribution functions of sequences in [0, 1] 2 and various extremal problems related to distribution functions.…”

On ideals defined by asymptotic distribution functions of ratio block sequences

“…. The set of distribution functions of ratio block sequences was studied in [1][2][3][4][5][6][7][10][11][12][13][14].…”

On structure of the family of regularly distributed sets with respect to the union

“…. The set of distribution functions of ratio block sequences was studied in [1][2][3][4][5][6][7][9][10][11][12].…”